Effect of indirect exchange interactions on the magnetic properties of europium monochalcogenides

Physica 95B (1978) 11-22 0 North-Holland Publishing Company

EFFECT OF INDIRECT EXCHANGE INTERACTIONS

ON THE MAGNETIC PROPERTIES OF

EUROPIUM MONOCHALCOGENIDES G. ter MATEN and L. JANSEN Institute of Theoretical Chemistry, Universityof Amsterdam, Amsterdam, The Netherlands Received 23 January 1978 Revised 31 March 1978

Magnetic structures and transition temperatures of the solid Eu-monochalcogenides (EuO, EuS, EuSe and EuTe) at low temperatures are analyzed on the basis of exchange perturbation theory. An effective-electron model is used, with one electron per Eu-cation, two spin-paired electrons per anion. In the analysis, up to three anions per exchange unit are considered. Rather good agreement with experiment (types of ordering, transition temperatures) is obtained within a certain range of effective-electron parameters. It is found that the extension of the effective orbital on Eu2+must be large to also account implicitly for the effect of its valence-shell electrons. However, the accompanying orbital for 02- is much too small to be acceptable. The extension to two- and three-anion exchange units does not remove this anomaly. It is concluded that the valence shells of Eu2+ must be explicitly taken into account in consistently explaining the magnetic properties of these solids.

Eu-perovskites such as EuLiI-I, (Tc = 37.6 K). The remaining magnetically ordered rare-earth compounds are either antiferromagnetic, or they exhibit more complex magnetic structures. For a survey, we refer to a review article by Methfessel and Mattis [6]. The explanation of these phenomena is as yet far from complete. It is certain, as in the Mn-series, that exchange interactions are responsible for the magnetic structure; dipole-dipole interactions, crystal-field effects and spin-orbit interactions are much too weak to account for the relatively high transition temperatures. Only in a case such as terbium ethyl sulfate, ferromagnetic with Tc = 0.24 K, can the magnetic behaviour be simply explained in terms of a magnetic dipole model [7]. It is generally accepted that magnetic ordering in .3d-solids such as the Mn-chalcogenides is caused by two types of exchange interaction: direct exchange between the paramagnetic cations, and indirect exchange (“superexchange”) involving at least one anion of the lattice. The history of superexchange theories, starting with Kramers [8] in 1934, is too wellknown to be given here. We refer to the literature for excellent detailed reviews [9], A recent discussion

1. Introduction The discovery of ferromagnetic EuO (Tc = 69.4 K) in 1961 by Matthias, Bozorth and Van Vleck [l] aroused intensive and lasting interest in the study of magnetic properties of the mdnochalcogenides EuO, EuS, EuSe and EuTe. It is now known that also EuS is ferromagnetic (Tc = 16.5 K) [2] and that EuSe is antiferromagnetic, ferrimagnetic and again antiferromagnetic in the temperature region from 0 K to 4.6 K [3,4] ; EuTe is definitely antiferromagnetically ordered below TN = 9.6 K [5] . The behaviour of these solids deviates, therefore, strikingly from that of the corresponding Mn-series: Mn-chalcogenides with the same Bl (NaCl) structure as the Eu-salts (i.e. MnO, MnS and MnSe; MnTe has the B8 (NiAs) structure) are all antiferromagnets wrth NCel temperatures between 100 and 200 K. We note that Eu2+ and Mn2+ have spherically symmetric ground states 8S7,2 and $12, respectively. Still, ferromagnetism in solids of rare-earth compounds is the exception rather than the rule; of binary compounds only EuO, EuS and GdN (Tc = 60 K) are ferromagnets. A few further examples are found with 11

12

G. ter Maten and L. Jansenhdirect

exchange interactions in Eu monochakogenides

of the state of first-principle superexchange theory has been given by Newman [lo] . A type of theoretical approach to magnetic properties of Eu-salts similar to that followed for 3d-solids meets with serious obstacles. A major complication lies in the fact that the 4f-shell of Eu2+ is deeply imbedded in the valence shells of that cation, in contrast with the 3d-shell of Mn2+. As a result, overlap of 4f-wavefunctions on different Eu2+-ions must be very small and cannot be expected to explain the observed transition temperatures. Since EuO is a non-conductor at the temperatures involved, indirect exchange via conduction electrons must be excluded. Various suggestions regarding the role of the Euzi valence electrons in cation-cation exchange have been made. Trammel [ 111 supposes that these interactions must occur via the valence electrons, without specifying the mechanism. Callaway [ 121, and De Graaf and StrLsler [ 131, considered exchange via filled valence bands. Kasuya [ 141 suggested that the Sdwavefunctions of Eu2+, being more extended than the 4f-functions, overlap sufficiently to form an empty 5dconduction band. He supposed that nearestneighbour Eu2+ exchange proceeds mainly indirectly in that a 4f-electron is virtually transferred to the vacant Sd-band, couples ferromagnetically via d-f exchange to a 4f-electron at the second Eu2+, and then returns to its initial 4f-state. In earlier analyses [15, 161 we have shown that a quantitative understanding of the magnetic ordering in Mn-chalcogenides can be obtained on the basis of an indirect-exchange unit consisting of two cations and one anion; this concept dates back to Kramers (1934) [S] . In the model we use, the five unpaired 3d-electrons of each Mn2+ are replaced by one “effective” electron, those of a closed-shell anion (02-, S2-, etc.) by two spinpaired effective electrons. Since both for Mn2+ and the anions the total orbital angular momentum is zero, we choose orbitals of spherical symmetry for the effective electrons. To this unit (four electrons on three centers), we applied a Rayleigh-Schrbdinger type of exchange perturbation theory [ 171, taking into account full permutation symmetry of the electrons, and starting from free-ion wavefunctions in zeroth order. Recently, the model was also applied to an analysis of magnetic structures of the manganese pyrites MnS,, MnSe2 and MnTe, [ 181, which deviate from “normal” (AF2) behaviour

in that their structures are AF3, AF3/AFl and AFl, respectively. The question which we will attempt to answer in this analysis is whether or not such a model may also be developed for 4f-solids, in particular for those of the Eu-monochalcogenides. Since the 4f-shell of Eu2+ is deeply imbedded in the valence shells, an obvious extension would be to add a shell of two spin-paired effective electrons to represent these valence shells. The maximum of the charge distribution for this shell must then lie outside that of the shell representing the 4f-electrons. Such a description leads, for a Euanion-Eu unit, to a three-center, eight-electron model, comprising an indirect valence-shell contribution to the direct and indirect cation-cation interactions. Preliminary calculations on this basis have been carried through [15, 191. However, the presence of two shells of effective electrons on the same center was found to imply a considerable loss of transparency in the interpretation of the results. In view of this difficulty, it is more straightforward to investigate under what conditions, if any, the simple superexchange model as applied to 3d-solids can yield results covering also the magnetic properties of Err-monochalcogenides. If this should be possible, then such a model incorporates implicitly the effect of the valence electrons of Eu2+. A necessary condition is, obviously, that the anion parameters must be the same as for the Mn-chalcogenides. The following analysis deals with this subject,

2. Magnetic properties and coupling constants There are good reasons to assume that a simple Heisenberg Hamiltonian (i.e. isotropic and bilinear) -2JAB SA - SB for two cations A and B is a valid approximation to the description of magnetic properties of the Eu-salts under study. Here, S, and Sn are the total-spin angular momentum operators for the cations and JAB is the exchange coefficient. The magnetic molar crystal energy, in terms of spin operators, is then of the form (N is Avogadro’s number) -N

2

JABSA*SB.

B#A

For an arbitrary pair of cations, JAB can formally be written as the sum of direct exchange Jfd (no anion

G. ter Maten and L. Jansenllndirect exchange interactions in Eu monochalcogenides

involved) and a contribution via the anions of the lattice (indirect exchange), i.e. J AB = Jfd + 2 J_$) + (terms involving more than one anion), m (2) with Ji’) the indirect exchange coefficient involving anion m. If all cation sites are equivalent, as in the present case, then we replace JAB by J, if B is a nearest neighbour of A, by J2 if B is a next-nearest neighbour, etc. It is often assumed that interactions beyond J2 may be neglected (see, however, below). Solving the so-called molecular field equations yields three antiferromagnetic spin alignments AFl, AF2 and AF3, plus the ferromagnetic ordering F. From these equations, one readily derives two conditions for stability of the ferromagnetic alignment, namely J, > 0 and Jz > -J,, whereas the AF2 structure (observed with EuTe and practically all nonconducting 3d-solids) is the most stable under the conditions J2 < 0 and 252 < J1 < -J,. The stability conditions for the structures AFl and AF3 (not observed with the Eu-monochalcogenides) are AFl : Jl0;AF3:Jr
IJ11/lJ21 > 2. In the literature [6] it is supposed that the predominant contribution to J2 arises from a linear Euanion-Eu array, so-called “180’ superexchange”, measuring the angle at the anion. It is also generally assumed [6,20] that J, is predominantly to be ascribed to direct cation-cation exchange. Going from EuO to EuTe, J, will then rapidly decrease as the cation distances become larger. This decrease would be slower for J2, since at the same time the anion becomes larger, counteracting the effect of increase in the Eu-Eu separation. If it is further assumed that J, is positive over the whole range, and that J2 is always negative, then the transition from ferromagnetic to antiferromagnetic behaviour can at least qualitatively be explained. It should be noted that such a simple interpretation is not imposed by the stability conditions. Specifically, J2 could be positive for EuO, and Jr need not be positive for EuTe to render the observed structures stable. Next, we briefly survey the data on coupling constants Jn as deduced from experiments. This is relevant

13

for what follows, also since different methods used in interpreting the data yield in part considerable differences between these values. The experimental sources are: (a) inelastic neutron scattering methods; (b) hypertine interactions (NMR and Mijssbauer techniques). ad (a) Earlier experimental difficulties with neutron scattering experiments, due to strong absorption of neutrons by the Eu2+ -cations, appear now to have been overcome. By using polycrystalline powder samples, Passe11 et al. [20] measured the spin-wave spectrum in EuO and EuS over the entire Brillouin ’ zone. They employed a two-parameter model in interpreting the data and determined J, and J2 separately. It should here be noted that, if exchange between more-distant cations cannot be neglected, J, and J2 thus determined represent only effective coupling constants. ad (b) By analyzing the transferred hyperfine fields at Eu nuclei in EuX solids (X = 0, S, Se, Te), Zinn et al. [21] found evidence for the existence of a longrange exchange mechanism. Although the coupling constants (J,) themselves could not be evaluated because of an unknown relation with the hyperfinefield terms (Bn), they showed that using only two terms (B1 and B2) did not explain the experimental results. The discrepancies could be removed by adding higher terms (up to at least &). These authors also question the way Passe11 et al. applied spin-wave theory. As a summary we have collected relevant experimental data on the Eu-monochalcogenides in tables I, II and III below. In table I are listed the nearestneighbour cation-anion and cation-cation distances, the measured paramagnetic Curie temperatures 0, the ordering temperatures and types of magnetic ordering. These data were taken from ref. 21a. In table II we list values of J,/k and J2/k frequently quoted in the literature [5], together with the calculated paramagnetic Curie temperatures 8. Finally, in table III we list the corresponding “best” values of J,/k and JZ/k proposed by Zinn [2 1a] . Note that in table III, J2 for EuO has the positive sign, in accordance with the results by Passell et al. [20]. This sign agrees with theoretical predictions by Kasuya [ 141, and with empirical results by Swendsen [22] based on the “Modified Callen Decoupling” (MCD) approximation of the Heisenberg model. The sign

14

G. ter Maten and L. Jansenflndirect exchange interactions in Eu monochalcogenides

Table 1 Relevant experimental

data on the solid Eu-monochalcogenides

[crystal structure Bl (NaCl)] . Data taken from ref. 2la.

Compound

Cation-anion distance (au)

Cation-cation distance (au)

Baramagnetic Curie temperature (K)

Ordering temperature (KI

Kind of magnetic ordering

EuO EuS EuSe EuTe

4.86 5.64 5.85 6.23

6.87 7.97 8.28 8.82

78 19 8.5 -4

69.4 16.5 4.6 9.6

F F complex* AF2

+4 ?z5 f 5 i2

i f * i

2 0.2 0.05 0.1

* See refs. 3 and 4.

Table II Experimental coupling constants Jl/k and Jz/k collected by Wachter [ 51, and paramagnetic Curie temperatures 0 using these coupling constants. Except for EuS, no experimental errors are given in the literature. Compound

Jlfk W

Jzh W

e W

EuO EuS EuSe EuTe

0.63 0.20 f 0.01 0.13 0.03

-0.07 -0.08 f 0.02 -0.11 -0.15

75 20 f 3 9.5 -5.7

Table III “Best” coupling constants Jl/k and Jz/k collected by Zinn ]2la], using more recent data compared to table II, and paramagnetic Curie temperature 0 using these collected coupling constants. Compound

J& W

EuO EuS EuSe EuTe

0.55 0.21 0.11 0.06

J# (K) f 0.05 f 0.03 f 0.01 * 0.02

+0.15 -0.11 -0.09 -0.20

f f i f

0.1 0.01 0.01 0.05

79 19.5 8.2 -5.0

+13 f 4 f 2 *6

does, however, not agree with the theoretical result obtained by Falkovskaya and Sapozhnikov [23]. The conclusion is that there exists considerable uncertainty concerning “experimental” values of the coupling constants. Theoretically, very little information is available in support of one or the other set of data. There is another point to be noted here. Taking either set of values for J1 and J2 (table II or III) there is little difference between the slopes of the Jr- and

J2-curves as a function of the Eu-Eu distance. Provided the interpretation of the data in terms of two coupling parameters is valid, the small difference in slope between the two curves indicates that the contribution to J, from indirect exchange (90” superexchange) must be considerable, contrary to what is commonly assumed. Conversely, this would qualitatively explain the apparent discrepancy, observed by Pas@ et al. [20], and earlier by McWhan et al. 1241, in the values of d(lnT&/d(lnQ obtained from measurements of JI and 52 for EuO and EuS on one hand, and direct measurements of the pressure dependence of Tc for EuO and EuS on the other hand. The reason is that, if indirect exchange interactions contribute significantly to J, (and J,), then decreasing the distance between the cations through substitution of a smaller anion is not the same as applying the pressure necessary to produce the same compression, while leaving the anion unchanged. In the former case, Tc will increase less rapidly with decreasing volume than in the latter case, in agreement with experiment. We will return to this point later on. Assuming now that an analysis in terms of J, and J2 is a valid basis for the analysis, it is the task of theory to account for the changes in J, and J2, going from EuO to EuTe. To our knowledge, the only numerical many-electron analysis which has been carried through is that for J2 of EuO by Falkovskaya and Sapozhnikov [23], using a method of configuration interaction applied to the unit Eu-0-Eu. They obtained a (negative) value whose magnitude is in reasonable agreement with experiment. The exchange constant J, was not evaluated, so that no conclusion could be drawn as to the stable magnetic structure of EuO.

G. ter Maten and L. Jansen/Indirect

3. The effective electron model. Formalism For the evaluation of the coupling constants, we will adopt the effective-electron model already mentioned in the Introduction. The simplest exchange unit consists of two cations A and B, and one anion C. The four effective electrons (one per cation, two on the anion) are labelled 1,2,3 and 4. The effect of the Eu-valence shells on the exchange interactions will only implicitly be taken into account through the size of the effective-electron orbital on the cations. The free-ion orbital wavefunctions (i.e. no interactions between the ions) are denoted by QA, ap, for the cations and by @‘cfor the anion. Note that these wavefunctions are not mutually orthogonal. In the application of the Rayleigh-Schrijdinger type exchange perturbation theory [ 171, we start from a zeroth-order wavefunction

By making use of a so-called “symmetric double-coset” (SDC) decomposition of permutation groups [25] with respect to the invariance group of the product function @‘,it is readily found that & can be written as a sum of different contributions of interatomic permutations, different for different spin states. Transforming the denominator in (4) in a similar way, the expression (4) becomes

(4’) For any given system of effective electrons, & can be determined for the different spin states. Since we here consider only two unpaired (effective) electrons, Eil) can always be written [26] as the expectation vaue of a bilinear Heisenberg Hamiltonian, C-

‘Zf==A@A(1)@~(2)@)C(3)@~(4)u =AQa,

(4) The perturbation V comprises Coulomb interactions between electrons at different centers, those terms between electrons at one center and the nuclei at the two other centers, as well as nucleus-nucleus interactions. Since V does not involve spin variables, the integration over spin space in (4) can be carried out at once. The numerator then takes the form VP@) (u,Pu) = ((a, V&J),

P

with & = 2 (-l)Q+J,Pu). P

2Js,

l

s2,

(3)

with A = Zp(-l)pP, the antisymmetrizer the total system and with u the well-known triplet or singlet eigenfunction(s) of the squared total-spin operator. The permutations P run over the symmetric group on four elements. (If one more anion is included, then \k in (3) is to be extended correspondingly.) The associated first-order energy .@’ is, in good approximation (only terms linear in the perturbation v) given by [ 171

1 (-l)P(@,

15

exchange interactions in Eu monochalcogenides

(5)

with s1 and s2 the spin angular momentum of the two electrons, and with

operators

j = @Al) - E$l))/2 (S stands for “singlet”, T for “triplet”). The quantity j, as calculated from the model, must now be compared with the coupling constant JAB for the cations. We suppose that j in good approximation represents the total exchange constant ~iJJii, where i runs over the unpaired electrons of cation A, j over those of B. For cations with half-filled shells (Mn2+, Eu2+, Gd3+), JAB = IZ;i,jJij/n2 [27] , with n the number of unpaired electrons per cation. One thus obtains JAB = j/n 2 = (E;l) - E3/2n

2.

(6)

This is the basic equation for a comparison between results of the effective-electron model and values of JAB deduced from experiments. For the orbital function QA(QB) occupied by the effective electron on each cation, we choose a spherically symmetric Slater function *p(T) = rp- l e- hrlaO, with r the distance to the nucleus and h a variable parameter. Since calculations of the splitting Eil) - Ey’ with p = 1,2,3,4 turn out to yield very similar results [ 151 , we limit ourselves to 1s-Slater

16

G. terhfaten and L. Jansen/lndirect exchange interactions in Eu monochalcogenides

functions (p = 1). Also, the two effective electrons on an anion occupy a Is-Slater orbital. Consequently, there are two quantites to be determined: X, (cation) and A, (anion). In earlier work on the crystal stability of ionic solids [28] and the magnetic properties of nonconducting solids of 3d-compounds [ 16, 191, the h-values for the anions 02-, S2-, Se2- and Te2were determined relative to one another through the ratios of their experimental diamagnetic susceptibilities (proportional to Xe2 in the model). This reduces the number of X-parameters to two. In later work [26, 291, only the contributions of the anion valence shells to the susceptibility were taken to determine the X-ratios. We will discuss the procedure in detail below. The A-value for Eu2+ will be left free in the analysis.

4. Evaluation of .I, and J2. Geometric configurations considered For any given values of the orbital parameters h,, h, for cation, anion, we can now determine, in first order of approximation, the coupling constants J, (nearest cation neighbours) and J2 (next-nearest neighbours), on the basis of eqs. (4’) and (6). From these results, the relative stability of different mag netic patterns at 0 K are evaluated. It should be noted that the paramagnetic Curie temperature 8 is a more reliable quantity to be compared with model calculations than the transition temperatures (NCel or Curie), since the molecular field approximation, on which eq. (3) is based, generally overestimates these temperatures [21a]. According to the definition of the coupling coefficient JAB (eq. 2), the basic unit needed for the calculations contains two cations plus all the anions giving a non-negligible contribution to the exchange interaction. In view of the short-range character of these interactions, we can severely limit the geometric configurations of anions around A and B as well as the number of anions to be considered. On the basis of trial calculations, the following indirect-exchange units were selected: Nearest-neighbour configurations (J1): (see fig. 1) Choosing the origin of our coordinate system at the center of the cube, we select the nearest-neighbour cation pair at (O,O,O); (fa,fa,O) with R AB = ia@. In

a

e

-J2

Fig. 1. Unit cell in the Bl (NaCl) structure of the Eu-monochalcogenides. Below the 90” superexchange (Jl) and 180” superexchange (J2) units are drawn.

descending order of importance, groups of anions: a)

(fa,O,O)(O,la,O);

0)

(+a,$a,$) (ia,4,-ia)

7)

(0,~440)

(da,O)

we then have three

(O,O,ta) (O,O,-:a); (-fa,O,O) (fa,a,O).

Other anions give negligible contributions to the exchange energy. Next-nearest neighbour configurations (52): (see fig. 1) Keeping cation A at the origin, we choose a nextnearest neighbour B at (0, a,O) with RAB = a. In descending order of importance, we then have two groups of anions: a’)

(O,ta,O);

0’)

(la,O,O) (-3a,O,O) (0,Oda) (O,O,-fa) (ia,a,O) (+,a,O)

(O,&)

(O,a,-fa).

The contribution by other anions was found to be negligible. As a first approximation, the coupling constants JAB will be evaluated, applying eq. (2), on the assumption that only one anion of the groups 01),fl), y) for J,, and CX’),0’) for J2 need be taken into account. From

G. terhfaten and L. Jansenjlndirect exchange interactions in Eu monochalcogenides

17

fig. 1, the following expressions for J, and J2 are then readily derived:

and five-atom components of J, and J2 must be added to the expressions given by eq. (7):

Jl = Jf”) + 2 Jr’

A2 J,

+ 4 510) + 4 Jp);

= JfW2)

+

4 Jl(0l.P)

+

4 Jfd;

(7)

Here, the superscript (0) again denotes “direct interaction”; the remaining superscripts denote one anion from the corresponding groups. Computationally, it was found that contributions to J, and J2 involving one anion result primarily from anions of group a) (so called “90’ superexchange”) and from group ar’) (“180’ superexchange”), respectively. The corresponding configurations are also given in fig. 1. The above equations apply for exchange involving only one anion in each cluster. That this is not always a valid approximation was recently found by Van Kalkeren et al. [ 181: the observed magnetic structures and transition temperatures of the manganese pyrites a second can only be accounted for b includin r and TeZ-) 4 in the “molecular anion” (S,“-, Se2exchange unit. Extending the formalism needed for the evaluation of two- and three-anion exchange is straightforward. The zeroth-order wavefunction (3) is again of the form A@J, where A is now the antisymmetrizer for a system of six (two cations and two anions) or eight (two cations and three anions) effective electrons, and where @ is the corresponding simple product of effective-electron wavefunctions. The coupling constants J (i.e. JI and Jz) are now obtained as a sum of contributions from zero, one, two and three anions, i.e.

J = J(o)

+

2 J(i) + 2 J&j) + i

i
1

J(',/, k),

i
with the superscript (0) denoting direct cation-cation exchange, and where (i), (i,j) and (i,j, k) denote the inclusion of one, two and three anions, respectively. It should be noted that, for example, J(‘*n refers to a two-anion contribution only, i.e. one-anion terms as well as the direct interactions between the cations have been subtracted. The different anions are chosen from among those of the groups a), p) and 7) for J,, from a’) and 0’) for Jz. Explicit evaluation shows that the following four-

A3J2=0.

(7’)

In the notation, A2 J and A3 J stand for two- and three-anion components, respectively, of J; cwland o2 are the two anions from the group cr), whereas a superscript cr, fl or 7 denotes any one anion from that group. Three-anion contributions to J2 were all found to be negligible.

5. Numerical results and discussion 5.1. Determination of the anion parameters h, As stated before, the h-values for 02-, S2-, Se2and Te2- are determined, relative to one another, through the ratios of their diamagnetic susceptibilities x. Specifically, since only the valence shell electrons must be expected to contribute significantly to the exchange interactions, we determine these ratios on the basis of the valence shelZpart, xvs, of x [26, 291. For two anions p and q, we then have the relation

(8) The contributions xvs were computed by using an atomic Self-Consistent Field programme of the Hartree-Fock-Slater type, adapted to the case of double-minus ions in a crystal [30]. Stable solutions were obtained by adding a positively charged sphere around the anion, with a radius equal to the so-called crystal radius of the ion [31]. The total charge on the sphere was varied until agreement with experimental x-values was reached [32]. Within experimental accuracy, this charge appeared to be the same for all four anions. From the SCF solutions, the quantity (r2) of the four valence shells was then evaluated, leading to the ratios h/h,. The results are collected in table IV.

G. ter Maten and L. Jansenllndirect

18

exchange interactions in Eu monochalcogenides

Table IV Anion data needed to obtain the ratios of the ¶meters (diamagnetic susceptibilities x in 10d cgs). Anion

02s2Se2Te2-

x experimental b,

X

(r2)

calculated

valenceshell

2.14 3.59 3.82 4.20

18 c, 38 48 70

18 38 48 69

3.12 6.38 1.49 9.51

a) Ref. [31]. b) Ref. [32]. c) Extrapolated

&ux

02P Se2Te2-

(au)

experimental

calculated from eq. (9)

4.86 5.64 5.85 6.23

4.85 5.64 5.86 6.23

Least-squares parameters (RE,,x

= a + b h02-/~~2-)

a = 3.01 f 0.03 (au) b = 1.84 f 0.02 (au)

It is interesting to note that the calculated ratios ~2_/~~_ (X = O,S,Se,Te) appear to be a linear function of the Eu-X nearest-neighbour distance, REUX, i.e.

R EuX = a + b (ho2-/A,24

(9)

A least-squares fit gave the results collected in table V. The relation (9) enables us to calculate J-values as a continuous function of distances and orbital parameters. 5.2. i?re two-parameter model We now have two parameters interpret

the magnetic

ordering

1 0.70 0.64 0.57

(see [32] ).

Table V Least-squares fit between nearest-neighbour cation-anion distances REuX and h-ratios (last column of table IV). Anion

hx2-/kf32-

Ionic radius au a)

at our disposal to in the Eu-chakogenides,

namely ho2- and &a+. It will first be investigated whether or not values for these two quantities can be found, on the basis of the model, which yield quanti-

tative agreement with experiment for all four chalcogenides. The first condition to be met is the occurrence of ferromagnetism with EuO, implying Jl > 0 (and J2 > -Jl). The model indeed predicts X-values for which J, > 0; thisregion is indicated in the following fig. 2. The two dotted straight lines border the middle section in which J, > 0 according to the model; the coordinates are ho2and AEu2+. Next, we determine those points (if any) in the middle section of fig. 2 for which the calculated J, agrees with the experimental value (table II). Again, the model does predict the existence of such h-values; the corresponding points lie on the solid line drawn in fig. 2. Third, we also require that the J, for EuTe be positive (from table II). This condition limits the possible h-values for 02- and Eu2+ still further, in the following way: The bottom line J, = 0 is found to follow the relation Ax2--/&2+ = 0.7 for all X (= O,S,Se,Te). Since, from table IV, we have ATe2-/h02= 0.57, one obtains ho2-/hEu2+

>

1.23

as a condition for positive J, of EuTe. This line is also drawn in fig. 2. Thus, only the upper part of the curve yielding correct values for J, of EuO is left as the allowed parameter range of Ao2- and hEu2+. A number of conclusions can be drawn from these results: (i) For EuO, the h-parameter of the cation is smaller than that of the anion, implying that the orbital

.

G. ter Maten and L. Jansenjlndirect exchange interactions in Eu monochalcogenides ’

.’ JlCO

AO*I,.*

,se

, , ,- ,+Q=l.*3

/( ,*0=0.93

,'

,'

,'

,'

I.. a5

.’ as

0.7

, 0.9

*'

,'

,'

,,'

,’

,\+o

,’ ,’ ,’ I

8’

Jl
,'

,

.

,

,

,

0.9

1.0

1.1

1.2

1.3

, LAEu*+

Fig. 2. Dependence of J1 for EuO on the parameters hE,,Z+ and h02-. The two dotted lines border the region where J1 > 0. Points on the solid line correspond to pairs of h-parameters for which J1 agrees with experiment (Jl/k = 0.63 K). The upper part of the sold line starting from the intersection with the line fragment Q = 1.23 yields positive values for Jl/k for EuTe. The pair of A-parameters determined by the intersection with the line fragment Q = 0.83 represents the “3d-limit”. (Q = h02-/hEu2+).

of Eu2+ in the model is “larger” than that of 02-, confirming a result already established in earlier work [ 191. This is not surprising since, by scaling to the experimental JI, the A-parameter implicitly includes also the influence of valence shell electrons on the exchange interactions. The neglect of the valence shell in Eu2+ is also apparent when we try to apply a type of relation between the h-values of cation and anion which was found to yield accurate results for 3d-solids, namely

[291

(10) where X3d is the part of the cation susceptibility due to the 3d-shell and where xvs is again the valence-shell contribution to the anion susceptibility. Substituting 4f for 3d in (10) yields, for EuO, that h02-/hEus+ = 0.83, much lower than the limit 1.23 required to render JI for EuTe just positive. With this low value of the ratio it appears impossible to stabilize the ferro-

19

magnetic structure of EuO with a paramagnetic Curie temperature somewhere near the observed value. (ii) in fig. 3, values for JI /k are given as a function of the nearest-neighbour cation-anion distance Ro, for a range of values of Ao2- and AEu2+, all chosen to yield the correct J,/k value for EuO. The two points marked with “V” represent the limit ho2-/hEu2+. = 0.83. Experimental values are marked with “0” and connected by the solid line. As is apparent from the figure, rather good agreement for the three other chalcogenides is obtained for h02-/hEu2+ in the neighbourhood of 1.2. Comparison of the calculated results with the second set of experimental values for JI /k (table III) yields a very similar pattern. In table VI, calculated and experimental values for J, /k are given. The calculations were carried out for a ratio ho2-/hEu2+ around 1.2. Each value of this ratio determines ho2- and hgu2+ separately (from fig. 2), taking the experimental J,/k for EuO as gauge value. Results for J,/k show that this quantity is much less sensitive than Jl/k with respect to a change in the A-ratio. Except for EuO, the calculated J2/k are always considerably more negative than the experimental values. Calculated paramagnetic Curie temperatures yield by far the best agreement with experiments for a ratio ho2-/hEu2+ of 1.23. These values are (in K) 74 (exp. 75) for EuO, 24 (exp. 20) for Eus, 8.5 (exp. 9.5) for EuSe and -12 (exp. -5.7) for EuTe. All-in-all, we may conclude from the above analysis that application of a 3dsuperexchange model to the interpretation of magnetic properties of Eumonochalcogenides yields results which are mutually consistent and in rather good agreement with experiments. No such agreement has so far been obtained on a different basis. It appears that the A-parameter of the Eu2+-ion must be chosen rather low (0.98) in order to incorporate the influence of the valence shells surrounding the 4f-shell. It is surprising to note that the agreement with experiments even extends, at least qualitatively, to the distance dependence of J, and J2. In fig. 4 we plot the calculated results for the whole range of cationanion distances R from EuO to EuTe. Assumed is here a dependence 1J/k I = Rmn, which relation has been found to hold for 3d-solids [29] with Mn-, Niand Co-ions. For EuO, an Rmn dependence with n = 18 f 6 fits the experimental data [24], whereas

G. ter Maten and L. Jansenjlndirect exchange interactions in Eu monochalcogenides

20

Jllk

to-

t 0.9 0.8- -

6.0

-

6.2

Rc-a

Fig. 3. Values for Jl/k are given as a function of the nearest-neighbour cation-anion distance, for a range of values of ko2- and A,&?+, all chosen to yield the correct Jl/k value for EuO. The solid line connects the experimental points marked by 0. The points marked ., *, ‘A, 0, +, V correspond to the following values of,?=,, Z+and A$-: eO.87, 1.30; * 0.95, 1.24;A0.99, 1.19;~ 1.03, 1.13; 2+= 0.83. Jl/k in units K; + 1.06, 1.06;01.08,0.89.jThe twopointsmarkedwith “3$ represent the “3d-limit” h&/+,, R c_a in au.

Table VI Calculated and experimental Jl/k-values for different anions and different values of Q (= ,I 2-/hEu?+). A gauge value is chosen for h02- by fitting Jr ?k (exp) of EuO.

Q

ho2-

1.1 1.2 1.23 1.3

1.13 1.19 1.20 1.24

‘Eu’+ 1.03 0.99 0.98 0.95

Anion Jl/k calculated (K) Q= 1.1 1.2 02S2Se2Te2-

0.14 0.02 -0.13

1.23 1.3

Jl/k experimental (K) (same as table II)

calibration 0.63 0.25 0.30 0.38 0.20 0.13 0.17 0.25 0.13 -0.02 0.00 0.07 0.03

n a 16 is suggested in the case of EuS [33] . Later

measurements [34] yielded n = 27. However, a striking anomaly appears when we compare the best value for the h-parameter of the

02--ion’(l .20) with the one obtained in earlier work [35] on 3d-solids (0.90). The latter value agrees qualitatively with what one must expect on the basis of Hartree-Fock (HF) wavefunctions for this anion: a doubly-occupied effective-electron orbital, replacing the complete valence shell, should have a larger extension (lower X) than the HF-value. Defining this extension by (r2) = 3ao2/A2 HF,we find that for 02-, hHF = 0.98. This value is higher than the one obtained from the analysis of 3d-solids, as it should be, but much lower than the one obtained in the present analysis of Eu-monochalcogenides. Apart from the fact that the latter result is in itself unacceptable, we would be forced to conclude that the extension of the effective-electron orbital of one and the same anion changes drastically in passing from 3d- to 4f-solids. This would take away the basis from under the effective-electron model. Before reaching a final conclusion, we must still verify that the inclusion of more than one anion in the exchange unit does not significantly modify the results obtained so far. Detailed calculations [based

21

G. ter Maten and L. Jansenjlndirect exchange interactions in Eu monochalcogenides

d InlJl dlnlRl=-”

Q= 1.20

i 16-

J2

14-

1066420

0 4.6

1 5.0

52

I 5.4

S

11 5.6

Se

Te

I

1 5.6

6.0

6.2 _i

6.4

Rc-a

Fig. 4. Variation of the exponent n = -dlnIJl/dlnIR I for J1 and J2 with respect to the nearest-neighbour cation-anion R C_a. Note that different nearest-neighbour distances imply different k-values of the anion. R,_, in au.

on eq. (7’)] were carried out for two and three anions. The results can be summarized as follows: a moreanion exchange model, with adjusted orbital parameters for Eu2+ and Te2-, leads to the same conclusions as its one-anion version. Again, Eu2+ is a quite extended cation, and the h-values for the anions must be chosen abnormally large. Consequently, the conclusion must be that, in spite of the rather good agreement obtained, a 3dsuperexchange model, based on an effective-electron approximation, cannot be applied to the analysis of magnetic properties of the Eu-monochalcogenides. In a following publication we will show how the role of the valence shells can be explicitly taken into account.

References [l]

B. T. Matthias, R. M. Bozorth and J. H. van Vleck, Phys. Rev. Letters 7 (1961) 160. [2] P. Heller and G. Benedek, Phys. Rev. Letters 14 (1965) [3] ;lSchwob, Phys. Kondens. Materie. 10 (1969) 186. C. Kuznia and G. Kneer, Phys. Lett. 27A (1968) 664.

distance

141 R. Griessen, M. Landolt and H. R. Ott, Solid State Commun. 9 (1971) 2219. 151 P. Wachter, CRC Critical Reviews in Solid State Sciences 3 (1972) 189. [61 S. Methfessel and D. C. Mattis, in Encyclopedia of Physics, S. Fliigge, ed. (Springer, Berlin, 1968), Vol. XVIII/l, p. 359. f71 J. Felsteiner and Z. Friedman, Phys. Rev. B7 (1973) 1078. S. K. Misra, Phys. Rev. B 13 (1976) 2241. PI H. A. Kramers, Physica 1 (1934) 182. PI P. W. Anderson, Sol. State Phys. 14 (1963) 99. C. Herring, Magnetism IIB (1966) 1. [lOI D. J. Newman, Physica 86-88B (1977) 1018. 1111 G. T. Trammel, Phys. Rev. 131 (1963) 932. 1121 J. Callaway, Nuovo Cimento 26 (1962) 625. 1131 A. M. de Graaf and S. Striissler, Phys. Kondens. Materie. 1(1963) 13. 1141 T. Kasuya, IBM J. Res. Dev. 14 (1970) 214. 1151 E. Lombardi, G. Tarantini, R. Block, R. Ro&l, G. ter Maten, L. Jansen and R. Ritter, Chem. Phys. Letters 12 (1972) 534. [W L. Jansen, R. Ritter and E. Lombardi, Physica 71 (1973) 425. [I71 L. Jansen, Phys. Rev. 162 (1967) 63. W. Byers Brown, Chem. Phys. Letters 2 (1968) 105. D. S. Farberov, V. Ya. Mitrofanov and A. N. Men, Int. J. Quant. Chem. 6 (1972) 1057.

22

G. ter Maten and L. Jansenflndirect exchange interactions in Eu monochalcogenides

[ 181 G. van Kalkeren, R. Block and L. Jansen, Physica 85B (1977) 259, 93B (1978) 195. [ 191 R. Ritter, L. Jansen and E. Lombardi, Phys. Rev. B8 (1973) 2139. [ 201 L. Passell, 0. W. Dietrich and J. Als-Nielsen, Phys. Rev. B14 (1976) 4897. [21a] W. Zinn, J. Magn. Magn. Mater. 3 (1976) 23. [21b] Ch. Sauer and W. Zinn, J. Magn. Magn. Mater. 3 (1976) 46. C. Crecelius, H. Malletta, H. Pink and W. Zinn, ibid 5 (1977) 150. Ch. Sauer and W. Zinn, Physica 86-88B (1977) 1031. [22j R. H. Swendsen, Phys. Rev. B5 (1972) 116. [23] L. D. Falkovskaya and V. A. Sapozhnikov, Phys. Stat. Sol. (b) 71 (1975) 469. [24] D. B. McWhan, P. C. Souers and G. Jura, Phys. Rev. 143 (1966) 385. [ 251 See e.g. P. Kramer and T. H. Se&man, Nuclear Phys. Al36 (1969)45. Al86 (1972) 49. R. Block, Physica 70 (1973) 397,73 (1974) 312. [26] L. Jansen and R. Block, Physica 86-88B (1977) 1012. R. Block and L. Jansen, in Quantum Science (Plenum, New York, 1976) p. 123. L. Jansen and R. Block, Angew. Chem. Int. Ed. Engl. 16 (1977) 294.

[27] J. H. van Vleck, Mat. Fisica Teo 14 (1962) 189.

[ 281 See e.g. L. Jansen and E. Lombardi, Disc. Faraday Sot.

[29] [30] [ 311

[32]

[33] [34] [35]

40 (1965) 78. L. Jansen, in Crystal Structure and Chemical Bonding in Inorganic Chemistry, C. J. M. Rooymans and A. Rabenau, eds. (North-Holland Publ. Co., Amsterdam, 1975) p. 205. L. J. de Jongh and R. Block, Physica 79B (1975) 568. R. E. Watson, Phys. Rev. 111 (1958) 1108. L. Pauling, in The Nature of the Chemical Bond (Cornell Univ. Press, New York, 1960) p. 514. J. A. A. Ketelaar, in Chemical Constitution (Elsevier, Amsterdam, 1960) p. 28. W. Klemm, Z. Anorg. Allgem. Chem. 244 (1940) 377. The x-value for 02- was extrapolated from a comparison of values for all four ions in the gas phase and for those of crystals except 02-. P. Schwab and 0. Vogt, Phys. Letters 24A (1967) 242. Y. Hidaka, J. Sci. Hiroshima Univ. A35 (1971) 93. R. Block (private communication).